Circulant matrices of any order

Creates and tests for circulant matrices of any order

Usage

circulant(vec,doseq=TRUE)
is.circulant(m,dir=rep(1,length(dim(m))))

Arguments

vec,doseq

In circulant(), vector of elements of the first row. If vec is of length one, and doseq is TRUE, then interpret vec as the order of the matrix and return a circulant with first row seq_len(vec)

m

In is.circulant(), matrix to be tested for circulantism

dir

In is.circulant(), the direction of the diagonal. In a matrix, the default value (c(1,1)) traces the major diagonals

Details

A matrix \(a\) is circulant if all major diagonals, including broken diagonals, are uniform; ie if \(a_{ij}=a_{kl}\) when \(i-j=k-l\) (modulo \(n\)). The standard values to use give 1:n for the top row.

In function is.circulant(), for arbitrary dimensional arrays, the default value for dir checks that a[v]==a[v+rep(1,d)]==...==a[v+rep((n-1),d)] for all v (that is, following lines parallel to the major diagonal); indices are passed through process().

For general dir, function is.circulant() checks that a[v]==a[v+dir]==a[v+2*dir]==...==a[v+(n-1)*d] for all v.

A Toeplitz matrix is one in which a[i,j]=a[i',j'] whenever |i-j|=|i'-j'|. See function toeplitz() of the stats package for details.

If the elements of vec are distinct, circulant() will return a latin square. Function latin() is a synonym for circulant(), see latin.Rd.

References

Arthur T. Benjamin and K. Yasuda. Magic “Squares” Indeed!, American Mathematical Monthly, vol 106(2), pp152-156, Feb 1999

Author

Robin K. S. Hankin

See also

latin

Examples

circulant(5)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    2    3    4    5
#> [2,]    5    1    2    3    4
#> [3,]    4    5    1    2    3
#> [4,]    3    4    5    1    2
#> [5,]    2    3    4    5    1
circulant(2^(0:4))
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    2    4    8   16
#> [2,]   16    1    2    4    8
#> [3,]    8   16    1    2    4
#> [4,]    4    8   16    1    2
#> [5,]    2    4    8   16    1
is.circulant(circulant(5))
#> [1] TRUE

 a <- outer(1:3,1:3,"+")%%3
 is.circulant(a)
#> [1] FALSE
 is.circulant(a,c(1,2))
#> [1] TRUE

 is.circulant(array(c(1:4,4:1),rep(2,3)))
#> [1] TRUE

 is.circulant(magic(5)%%5,c(1,-2))
#> [1] FALSE